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Analytic weak-signal approximation of the Bayes factor for continuous gravitational waves

Reinhard Prix

Abstract

We generalize the targeted $\mathcal{B}$-statistic for continuous gravitational waves by modeling the $h_0$-prior as a half-Gaussian distribution with scale parameter $H$. This approach retains analytic tractability for two of the four amplitude marginalization integrals and recovers the standard $\mathcal{B}$-statistic in the strong-signal limit ($H\rightarrow\infty$). By Taylor-expanding the weak-signal regime ($H\rightarrow0$), the new prior enables fully analytic amplitude marginalization, resulting in a simple, explicit statistic that is as computationally efficient as the maximum-likelihood $\mathcal{F}$-statistic, but significantly more robust. Numerical tests show that for day-long coherent searches, the weak-signal Bayes factor achieves sensitivities comparable to the $\mathcal{F}$-statistic, though marginally lower than the standard $\mathcal{B}$-statistic (and the Bero-Whelan approximation). In semi-coherent searches over short (compared to a day) segments, this approximation matches or outperforms the weighted dominant-response $\mathcal{F}_{\mathrm{ABw}}$-statistic and returns to the sensitivity of the (weighted) $\mathcal{F}_{\mathrm{w}}$-statistic for longer segments. Overall the new Bayes-factor approximation demonstrates state-of-the-art or improved sensitivity across a wide range of segment lengths we tested (from 900s to 10days).

Analytic weak-signal approximation of the Bayes factor for continuous gravitational waves

Abstract

We generalize the targeted -statistic for continuous gravitational waves by modeling the -prior as a half-Gaussian distribution with scale parameter . This approach retains analytic tractability for two of the four amplitude marginalization integrals and recovers the standard -statistic in the strong-signal limit (). By Taylor-expanding the weak-signal regime (), the new prior enables fully analytic amplitude marginalization, resulting in a simple, explicit statistic that is as computationally efficient as the maximum-likelihood -statistic, but significantly more robust. Numerical tests show that for day-long coherent searches, the weak-signal Bayes factor achieves sensitivities comparable to the -statistic, though marginally lower than the standard -statistic (and the Bero-Whelan approximation). In semi-coherent searches over short (compared to a day) segments, this approximation matches or outperforms the weighted dominant-response -statistic and returns to the sensitivity of the (weighted) -statistic for longer segments. Overall the new Bayes-factor approximation demonstrates state-of-the-art or improved sensitivity across a wide range of segment lengths we tested (from 900s to 10days).
Paper Structure (34 sections, 113 equations, 4 figures)

This paper contains 34 sections, 113 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The $\mathcal{F}$-statistic formalism
  3. Signal model
  4. Likelihood ratio
  5. The $\mathcal{F}$-statistic
  6. Bayesian detection framework
  7. Bayes factor
  8. Rediscovering the $\mathcal{F}$-statistic
  9. Neyman-Pearson-Searle optimality
  10. Likelihood ratio in physical coordinates
  11. The $\mathcal{B}$-statistic
  12. The Bero-Whelan approximation $\mathcal{B}{_{\mathrm{BW}}}(x)$
  13. Generalized $\mathcal{B}$-statistic and weak-signal approximation
  14. Motivation
  15. New $h_0$ prior: introducing an amplitude scale $H$
  16. ...and 19 more sections

Figures (4)

  • Figure 1: Noise distribution of coherent $\beta$ for $T_{\mathrm{seg}}=900s$ (starting at GPS time $t_0=1234567890s$, data from H1,L1), for two sky positions $(\alpha,\delta)_1=(5.16,0.78)\,rad$, and $(\alpha,\delta)_2=(0.32,0.49)\,rad$. The corresponding weights are $(w^{+},w^{\times})_1=(0.02,0.23)$ and $(w^{+},w^{\times})_2=(0.01,0.82)$, respectively. Dashed lines show the theoretical generalized $\tilde{\chi}^2$-distribution of eq:98, and the histograms are computed on $1000000.0$ synthesized values for $\beta$.
  • Figure 2: Detection probability as a function of false-alarm probability for the three different coherent test cases (i)-(iii) previously considered in 2019CQGra..36a5013B. In all cases the signal population has a fixed relative amplitude of $h_{*}=10$.
  • Figure 3: Detection probability as a function of semi-coherent segment length $T_{\mathrm{seg}}$ at fixed total span of $T_{\mathrm{span}}=10d$ (with GPS start time $756950413s$), false-alarm probability $p_{\mathrm{fa}}=e-3$ and sky position $(\alpha,\delta)=(2,-0.5)\,rad$. The top panel is for a single detector (H1), middle panel is H1+L1 and bottom panel includes Virgo H1+L1+V1. The relative signal amplitude $h_{*}(T_{\mathrm{seg}})$ is adjusted for a fixed detection probability $p_{\mathrm{det}}(2\widehat{\mathcal{F}})=0.7$ for the standard semi-coherent $\widehat{\mathcal{F}}$-statistic.
  • Figure 4: Detection probability as a function of false-alarm probability for a semi-coherent search on three unequal-$\gamma_\ell$ segments of $T_{\mathrm{seg}}=25h$, with per-segment duty factors of $0.1$, $1$ and $1$, respectively, assuming two detectors H1 and L1. The signal sky-position is $(\alpha,\delta)=(2,-0.5)$ and the relative amplitude is fixed at $h_{*}=10$.